1. Strain dependence of the band structure and critical points of pseudomorphic Ge1-ySny alloys on Ge
Nalin Fernando,1 John Hart,2 Ryan Hickey,2 Ramsey Hazbun,2 James Kolodzey2, Stefan Zollner,1
1 Department of Physics, New Mexico State University, Las Cruces, NM
2 Department of Electrical and Computer Engineering, University of Delaware, Newark, DE

2. Compositional dependence of the lattice parameter aGe1-ySnyrel of relaxed Ge1-ySny alloys (solid) and out-of-plane lattice parameter a of pseudomorphic Ge1-ySny alloys grown on Ge (dashed), calculated from Vegard’s Law. Symbols show the out-of-plane lattice constant determined from (224) grazing exit reciprocal space maps and the relaxed lattice constant calculated from the Poisson ratio.

3. Compositional dependence of the in-plane strain ε||,out-of-plane strain ε, hydrostatic strain εH and shear strain εS for pseudomorphic Ge1-ySny alloys grown on Ge calculated using Eqs.\ (\ref{parl}),\ (\ref{perp}),\ (\ref{H_strain}), and\ (\ref{S_strain}). In-plane ($\circ$) and out-of-plane strain ($\square$) derived from x-ray diffraction data in Fig.\ \ref{lattice_a}

4. Energies of the three top valence bands at the $\Gamma$ point ($v_{1}, v_{2}, v_{3}$), the lowest conduction band at the $\Gamma$ point ($E_{c}^{\Gamma}$) and at the L point ($E_{c}^{L}$) in pseudomorphic Ge$_{1-y}$Sn$_{y}$ alloys grown on Ge as a function of Sn content calculated using Eqs.\ (\ref{v1}),\ (\ref{v2}),\ (\ref{v3}),\ (\ref{Ec_gamma}), and\ (\ref{Ec_L}). The reference energy (0 eV) was chosen as the valence band maximum at any composition.

5. Compositional dependence of the direct and indirect band gaps of pseudomorphic Ge$_{1-y}$Sn$_{y}$ alloys grown on Ge at 300 K, calculated from Eqs.\ (\ref{v2}),\ (\ref{Ec_gamma}), and\ (\ref{Ec_L}). +: Direct band gap of pseudomorphic Ge$_{1-y}$Sn$_{y}$ grown on relaxed Ge on Si from ellipsometry.

6. Real (dashed) and imaginary (solid) parts of the complex dielectric function of pseudomorphic Ge$_{1-y}$Sn$_{y}$ on Ge versus photon energy determined from ellipsometry.

7. Compositional dependence of the E$_{1}$ ($\circ$) and E$_{1}$+$\Delta_{1}$ ($\diamondsuit$) critical point energies of pseudomorphic Ge$_{1-y}$Sn$_{y}$ alloys grown on Ge from ellipsometry (derivative analysis). The solid lines are E$_{1}$ (black) and E$_{1}$+$\Delta_{1}$ (blue) for relaxed Ge$_{1-y}$Sn$_{y}$ alloys. The dashed lines are for E$_{1}$ (black) and E$_{1}$+$\Delta_{1}$ (blue) of pseudomorphically strained Ge$_{1-y}$Sn$_{y}$ alloys grown on Ge calculated using deformation potential theory.

8. Logarithmic intensity versus diffraction angle for the symmetric (004) $\omega-2\theta$ x-ray reflections of pseudomorphic Ge$_{0.89}$Sn$_{0.11}$ and Ge$_{0.904}$Sn$_{0.096}$ grown on Ge. The diffraction curves were vertically offset.

9. Numerical second-order derivatives of $\epsilon_{2}$ versus photon energy for: (a) Ge$_{0.904}$Sn$_{0.096}$ - 2 phase model, (b) Ge$_{0.904}$Sn$_{0.096}$ - 3 phase model, (c) Ge$_{0.89}$Sn$_{0.11}$ - 2 phase model and (d) Ge$_{0.89}$Sn$_{0.11}$ - 3 phase model. The solid lines represent a model fit with Eq.\ (\ref{criticalpoint}).

10. (a) High-resolution x-ray diffraction (224) grazing exit reciprocal space map of a Ge$_{0.904}$Sn$_{0.096}$ layer on Ge. Both the Ge substrate and the GeSn layer peaks have the same Q$_{||}$ indicating that the epitaxial GeSn layer is fully strained relative to the Ge substrate. (b) Atomic force microscopy image of the GeSn surface of the same sample showing an RMS roughness of 1.6 nm.